We finally got a chance to read Liping Ma's Knowing and Teaching Elementary Mathematics, and Wow, what a book! This book (Ma's PhD thesis) used interviews with U.S. and Chinese mathematics teachers to compare approaches and understanding about basic elementary mathematics: subtraction with regrouping, multidigit number multiplication, division by fractions, and explorations of new knowledge. Her findings have direct and practical implications for teachers, parents, and tutors.

Key Findings:

- U.S. teachers tend to focus on procedures rather than understanding the conceptual foundations of basic math operations(in one testing situation, 83% U.S., 14% Chinese used procedural knowledge only)

- Impaired conceptual knowledge impaired U.S. teachers' abilities to understand student mistakes and apply math reasoning to new problems or scenarios.

- U.S. teachers' use of manipulatives could be misapplied, for instance, removing the need for a child to understand regrouping, rather than illustrating principles regrouping or borrowing.

-U.S. teachers showed less flexibility and alternative problem solving than their Chinese counterparts.

-U.S. teachers seemed more likely to introduce entertaining or visually appealing aspects of math calculations, but sometimes at the expense of accurate conceptual teaching.

Example: "One thing I would do is....put either an apple, orange or whatever, in the spaces...I mean it could be some weird thing, even pictures of elephants. I do not care what it was. But the children memorized this and they said, oh I remember that [my teacher] said do not put anything there because that is where the orange was or that was the apple...Just put something different there so that it will hit their eye."

- Some Chinese teachers regularly showed a math problem with errors on the board, and challenging students to 'find the problem' and 'summarize the rule'. A discussion would then follow about the correct underlying concept.

- Only 43% U.S. math teachers arrived at the correct answer to: 1 3/4 divided by 1/2 compared to 100% of the Chinese teachers. Problems included over-reliance on a mneumonic (without conceptual understanding), a lack of alternative ways to solve the problems, and limited analogical understanding. For the latter, for instance, U.S. teachers tended to rely on pizza-type examples for fractions, but became stumped with dividing by the 1/2.

- Finally, U.S. teachers also fared poorly in a scenario of a student suggesting a novel theory that increasing the perimeter of a closed figure would necessarily increase the area. 9% accepted the theory without a doubt, 78% would not have conducted any mathematical investigation ("I'd try to look it up in a book"), and only 13% would have investigated the claim. Only 1 teacher out of the 23 was able to solve the answer. In comparison, of the Chinese teachers, 8% accepted the claim outright, 70% arrived at the correct answer, and the remaining at least tried with mathematical reasoning, but solved the problem incorrectly.

Clearly we have a long way to go with teaching conceptual foundations in math. Bravo, for the math teachers at NCTM for taking a hard look at the book, and considering reforms.

(HT:artofproblemsolving.com)

NCTM: Comparing U.S. and Chinese Elementary Math

So many thoughts...

ReplyDeleteFirst if you can't get that division then you should be no where near teaching math beyond counting. Imagine swim classes taught by people who could only move across a poll by continuously jumping off the bottom. How deep of water would you trust such a person with your child?

This study while great is typical study of what is, and not of what could be.

What is needed is an engineering approach to solving this very complicated problem. An approach that builds on answering a series of increasingly complex questions.

How do these differences relate to the outcomes of the children? Can the differences be broken don't their effects measured separately? Do different procedures work well with different children? Can those differences be predicted ahead of time? What combination of procedures should be used with different children. What series of combinations of procedures should we try based on the knowledge and learning style of the individual child.

Then to go beyond the basics:

How does the interest of the of the child affect the effectiveness of the method developed. What role should the child's interest and choice be given in the process.

The link to the art of problem solving can demonstrate just how far kids can go that are interested in taking the journey. Its just unspeakably cool that some of the top mathematical talent in the county is applying their efforts to help kids learn maths.

You raise several interesting points. It has been very helpful to have real-world mathematicians taking an interest in mathematics teaching at the elementary school level (they have kids too). But there would also be a lot of benefit having having more engineers voice their thoughts about education as well.

ReplyDeleteIf the conceptual foundations are shaky, then the building will fall. Ma's work struck a chord because it showed at even the simplest level, concepts were not understood. There is a great interesting in teaching children about applications of information they're learning, but it is important that we know that that the teachers know the concepts, before trying to discuss applications...otherwise they'll be stuck parroting the applications that others suggest rather than seeing connections on their own.